A Data-Driven Model of Portland Policing
2023 Poster Details & References
The Policing Mathematical Model
Variables
- \(S \longrightarrow\) (susceptible individuals) are people who are not currently offenders or victims of crime, but who are at risk of becoming either.
- \(P \longrightarrow\) (offenders or individuals who are allegedly committed crimes) are people who have committed a crime, whether or not they have been caught or convicted. This includes both victim-less crimes and crimes with victims.
- \(V \longrightarrow\) (victimized individuals) are people who have been the victim of a crime. This includes both physical and emotional crimes.
- \(O \longrightarrow\) (police presence, dispatches, or interactions) is the amount of police activity in a given area. This can include things like foot patrols, traffic stops, and arrests.
- \(C \longrightarrow\) (reported crimes after the police encounter) are crimes that have been reported to the police and have been investigated.
- \(t \longrightarrow\) (time where \(t \in [0,1]\) discretized evenly for 12 months) is the time in months, discretized evenly from 0 to 1. This means that there are 12 time steps, each representing one month.
Parameters
- \(\mu_s \in \mathbb{R^+} \longrightarrow\) (growth/decay rate of the susceptible) is the rate at which the number of susceptible individuals increases or decreases over time.
- \(\mu_p \in \mathbb{R^+} \longrightarrow\) (natural decay rate of the offenders) is the rate at which the number of offenders decreases over time due to natural causes, such as death or aging out of the criminal justice system.
- \(\mu_v \in \mathbb{R^+} \longrightarrow\) (natural decay rate of the victims) is the rate at which the number of victims decreases over time due to natural causes, such as death or recovery from the crime.
- \(\mu_o \in \mathbb{R^+} \longrightarrow\) (natural decay rate of police presence) is the rate at which the number of police officers decreases over time due to retirement, resignation, or death.
- \(\mu_c \in \mathbb{R^+} \longrightarrow\) (natural decay rate of crime reports) is the rate at which the number of crime reports decreases over time due to factors such as lost or misplaced reports, or reports that are never investigated.
- \(\alpha_s \in \mathbb{R^+} \longrightarrow\) (the rate at which susceptible become offenders) is the rate at which susceptible individuals are converted into offenders. This can happen through a variety of means, such as peer pressure, drug use, or exposure to violence.
- \(\alpha_{ss} \in \mathbb{R^+} \longrightarrow\) (the rate at which susceptible become offenders by interacting with another susceptible) is the rate at which susceptible individuals are converted into offenders through direct interaction with another susceptible individual. This can happen through a process of social learning, where one individual learns from another how to commit crimes.
- \(\alpha_{ps} \in \mathbb{R^+} \longrightarrow\) (the rate at which offenders interact with a susceptible - repeated consecutive offenses) is the rate at which offenders interact with a susceptible individual, resulting in the susceptible individual becoming an offender. This can happen through a process of coercion or intimidation, where the offender forces the susceptible individual to commit a crime.
- \(\beta_v \in \mathbb{R^+} \longrightarrow\) (the rate at which victims return back to susceptible status) is the rate at which victims of crime return to the status of susceptible individuals. This can happen through a variety of means, such as recovery from the crime, or moving to a new location.
- \(\beta_p \in \mathbb{R^+} \longrightarrow\) (the rate at which offenders return back to susceptible status) is the rate at which offenders return to the status of susceptible individuals. This can happen through a variety of means, such as rehabilitation, or aging out of the criminal justice system.
- \(\gamma_v \in \mathbb{R^+} \longrightarrow\) (the rate at which victims interact with the police that includes an encounter or submitting an online police report) is the rate at which victims of crime interact with the police. This can happen through a variety of means, such as reporting the crime to the police, or being contacted by the police as part of an investigation.
- \(\gamma_p \in \mathbb{R^+} \longrightarrow\) (the rate at which offenders interact with the police that includes an arrest or encounter) is the rate at which offenders interact with the police. This can happen through a variety of means, such as being arrested, or being stopped and questioned by the police.
- \(\delta_o \in \mathbb{R^+} \longrightarrow\) (the rate at which official crime reports are processed after an encounter with the police) is the rate at which official crime reports are processed by the police. This includes the time it takes to investigate the crime, and to file the report with the appropriate authorities.
System of Differential Equations
\[\begin{align} \frac{dS}{dt} = & \left( \mu_s - \alpha_s - \alpha_{ss} S - \alpha_{ps}P \right) S + \beta_v V + \beta_p P \\ \frac{dP}{dt} = & \left( \alpha_s + \alpha_{ss} S \right) S - \left( \beta_p + \mu_p \right) P \\ \frac{dV}{dt} = & \left( \alpha_{ss} S + \alpha_{ps} P \right) S - \left( \beta_v + \mu_v \right) V \\ \frac{dO}{dt} = & \gamma_v V + \gamma_p P - \left( \delta_o + \mu_o \right) O \\ \frac{dC}{dt} = & \delta_o O - \mu_c C \\ \end{align}\]
The system of DEs is solved using LSODE. LSODE, the Livermore Solver for Ordinary Differential Equations, is a robust numerical solver known for its efficiency in handling stiff systems of differential equations. Stiff systems of ODEs are characterized by rapidly changing solutions, making them challenging to solve with standard methods. LSODE excels in tackling these complex systems. When dealing with a stiff system of ODEs, selecting the right numerical solver, such as LSODE, can significantly improve the accuracy and efficiency of the computations. To accurately simulate the behavior of the system, the initial values for the ODEs are extracted from the data recorded at the system’s initial time point.
Fitting to Data
Maximum Likelihood Estimation (MLE) is used to fit model parameters into data under the assumption that the error terms are Poisson distributed.
The data in question is a collection of crime reports and police dispatches from the Portland Police Bureau open data source. Note that we are only fitting the variables crime reports (\(C\)) and the police presence (\(O\)).
Let \(\widehat{O}\) and \(\widehat{C}\) be the estimated values of crime reports and police presence, respectively, from the model. Let \(o\) and \(c\) be the actual values of crime reports and police presence, respectively, from the data.
The goal of MLE is to determine the parameters for which the observed data have the highest joint probability. First, we write the parameters of the model as a vector
\[\vec{\theta} = \begin{bmatrix} \mu_s & \mu_p & \mu_v & \mu_o & \mu_c & \alpha_s & \alpha_{ss} & \alpha_{ps} & \beta_v & \beta_p & \gamma_v & \gamma_p & \delta_o \end{bmatrix}^T.\]
So, we let the function \(\vec{f}(t;\vec{\theta})\) be the solution to the system of differential equations. Since we only fit the variables \(O\) and \(C\) then
\[\vec{f}(t;\vec{\theta}) = \begin{bmatrix} \widehat{O} & \widehat{C} \end{bmatrix}^T.\]
Next, let \(X\) be a Poisson random variable with probability density function
\[P(X = x;\lambda) = \frac{\lambda^x}{x!} e^{-\lambda}\]
where \(\vec{\lambda} = \vec{f}(t;\vec{\theta})\) and \(\vec{x} = \begin{bmatrix} o & c \end{bmatrix}^T\). So, the likelihood function is written as
\[L(\lambda_1, \cdots, \lambda_n; x_1, \cdots, x_n) = \prod_{i=1}^{n} \frac{\lambda_i^{x_i}}{x_i!} e^{-\lambda_i}\]
where \(n\) is the length of the vector \(\vec{x}\). So, the negative log-likelihood is written as
\[\begin{align} -\ln{L(\lambda_1, \cdots, \lambda_n; x_1, \cdots, x_n)} = & -\ln{\left(\prod_{i=1}^{n} \frac{\lambda_i^{x_i}}{x_i!} e^{-\lambda_i}\right)} \\ = & - \sum_{i=1}^n \ln{\left(\frac{\lambda_i^{x_i}}{x_i!} e^{-\lambda_i}\right)} \\ -\ln{L(\lambda_1, \cdots, \lambda_n; x_1, \cdots, x_n)} = & - \sum_{i=1}^n \left( x_i \ln{\left( \lambda_i \right)} - \ln{\left( x_i ! \right)} - \lambda_i \right). \end{align}\]
We seek to find model parameter values \(\vec{\theta}\) to minimize/maximize the negative log-likelihood function written above. We used the Nelder-Mead optimization method to find the set of parameter values that minimizes the negative log-likelihood function.
Estimated Parameters
Data and Model (Selected Neighborhoods)
Summarized Data (Selected Figures)
